How Do Molarity and Molality Depend on Temperature?

Molarity and molality both describe how much solute is present, but they use different denominators. That difference determines their usual response to temperature: a solution’s volume can expand or contract, while the masses of its solute and solvent do not change just because the temperature changes.

What molarity and molality measure

Molarity, also called amount concentration, is the amount of solute divided by the volume of the solution:

$$c = \frac{n_{\mathrm{solute}}}{V_{\mathrm{solution}}}$$

It is commonly expressed in moles per litre (mol/L, or M). The volume in the denominator is the volume of the whole solution, not just the solvent.

Molality is the amount of solute divided by the mass of the solvent:

$$b = \frac{n_{\mathrm{solute}}}{m_{\mathrm{solvent}}}$$

Its usual unit is moles per kilogram of solvent (mol/kg). The denominator is not the mass of the solution. This distinction matters when comparing concentrations or calculating changes with temperature.

Why temperature usually changes molarity

For a prepared sample in which no material enters or leaves and no reaction or precipitation changes the amount of dissolved solute, the number of moles of solute stays fixed. But the solution’s volume can change with temperature. Most liquids expand when heated and contract when cooled, so, in the usual case, warming lowers molarity and cooling raises it.

For the same sample at two temperatures,

$$c_2 = c_1\frac{V_1}{V_2}$$

where $V_1$ and $V_2$ are the solution volumes at the two temperatures. If the volume increases, the same amount of solute is spread through more volume, and the molarity decreases. The reverse happens if the volume decreases.

For a modest temperature change, the volume is sometimes approximated by $V_2 \approx V_1[1+\beta(T_2-T_1)]$, where $\beta$ is the solution’s volumetric expansion coefficient over that temperature interval. This is an approximation, not a universal correction: the coefficient depends on the solution and temperature, and volume changes need not be perfectly linear.

Why molality stays constant

If no material enters or leaves and the solution’s composition does not change, heating does not change the masses of the solute or solvent. Since molality uses the solvent’s mass rather than the solution’s volume, molality remains the same even if the solution expands.

This is why molality is often convenient when comparing properties measured at different temperatures. It avoids the direct temperature dependence caused by volume changes. It is not immune to changes in composition, however: if solvent evaporates, solute precipitates, or a reaction changes the amount of dissolved species, molality can change too.

Worked example

Suppose a sample contains $0.250$ mol of solute dissolved in $0.500$ kg of solvent. At $20\,^{\circ}\mathrm{C}$, its measured solution volume is $1.000$ L. After warming, the volume is measured as $1.012$ L. Assume there is no evaporation, reaction, or precipitation.

At the initial temperature, the molarity is

$$c_1 = \frac{0.250\ \mathrm{mol}}{1.000\ \mathrm{L}} = 0.250\ \mathrm{mol/L}.$$

At the warmer temperature, the amount of solute is still $0.250$ mol, but the solution occupies more volume:

$$c_2 = \frac{0.250\ \mathrm{mol}}{1.012\ \mathrm{L}} \approx 0.247\ \mathrm{mol/L}.$$

The molality at either temperature is

$$b = \frac{0.250\ \mathrm{mol}}{0.500\ \mathrm{kg}} = 0.500\ \mathrm{mol/kg}.$$

The example uses measured volumes; the size of the change is specific to the sample and temperatures. The general result is that volume expansion lowers molarity, while unchanged solute and solvent masses leave molality unchanged.

Converting between the two using density

Density can help convert between molality and molarity, but it must correspond to the solution at the temperature of interest. If $b$ is in mol/kg, the solute molar mass $M_s$ is in kg/mol, and solution density $\rho$ is in kg/L, then

$$c = \frac{b\rho}{1+bM_s}.$$

To see why, take $1$ kg of solvent. It contains $b$ moles of solute, so the solute mass is $bM_s$ kg and the total solution mass is $(1+bM_s)$ kg. Dividing that mass by the density gives the solution volume; dividing the moles of solute by that volume gives the formula above. Because density generally changes with temperature, this conversion can produce different molarities at different temperatures even when molality is fixed.

Common mistakes and limits

• Using solvent volume for molarity: Molarity uses the volume of the entire solution.
• Using solution mass for molality: Molality uses the mass of the solvent alone.
• Assuming molarity is unchanged because the amount of solute is fixed: Its denominator, solution volume, can change.
• Treating molality as unconditionally temperature-independent: It stays fixed only while the amounts of solute and solvent stay fixed. Evaporation, precipitation, or chemical change can alter them.
• Assuming every liquid expands in the same way: The direction and size of a volume change depend on the material and conditions. For precise work, use measured solution volume or density at the temperature being considered.